Sums of Power Sums

Let $f_k(n)$ be the sum of the $k$th powers of the first $n$ positive integers.
For example, $f_2(10) = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 = 385$.
Let $S_k(n)$ be the sum of $f_k(i)$ for $1 \le i \le n$. For example, $S_4(100) = 35375333830$.
What is $\sum (S_{10000}(10^{12}) \bmod p)$ over all primes $p$ between $2 \cdot 10^9$ and $2 \cdot 10^9 + 2000$?

To solve this problem, we can use nested loops to generate all possible area combinations for the four pieces of the triangle. We will iterate through each area, checking if the sum of the areas is less than or equal to n. If it is, we will add the area of the uncut triangle to the total sum.

Here’s the Python code to solve the problem:

“`python
def is_triangle(a, b, c):
# Check if three numbers can form a triangle
return a + b > c and a + c > b and b + c > a

def calculate_area(a, b, c):
# Calculate the area of a triangle given its side lengths (Heron’s formula)
s = (a + b + c) / 2
return (s * (s – a) * (s – b) * (s – c)) ** 0.5

def calculate_sum(n):
sum_area = 0

# Iterate through all possible area combinations
for a in range(1, n + 1):
for b in range(1, n – a + 1):
for c in range(b, n – a – b + 1):
d = n – a – b – c

# Check if the sum of the areas is less than or equal to n
if d >= 1 and is_triangle(a, b, c) and is_triangle(a, b, d) and is_triangle(a, c, d):
sum_area += calculate_area(a, b, c)

return sum_area

# Call the function to calculate S(10000)
result = calculate_sum(10000)
print(result)
“`

The code first defines three helper functions: `is_triangle` checks if three numbers can form a triangle using the triangle inequality theorem, `calculate_area` calculates the area of a triangle using Heron’s formula, and `calculate_sum` iterates through all possible area combinations and calculates the sum of the areas of the uncut triangles for valid quadruples.

Running the code will output the value of `S(10000)`, which is the sum of the areas of the uncut triangles represented by all valid quadruples with a total area less than or equal to 10000.

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